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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">ntv</journal-id><journal-title-group><journal-title xml:lang="ru">Научно-технический вестник информационных технологий, механики и оптики</journal-title><trans-title-group xml:lang="en"><trans-title>Scientific and Technical Journal of Information Technologies, Mechanics and Optics</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2226-1494</issn><issn pub-type="epub">2500-0373</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17586/2226-1494-2026-26-2-393-401</article-id><article-id custom-type="elpub" pub-id-type="custom">ntv-602</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>МАТЕМАТИЧЕСКОЕ И КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>MODELING AND SIMULATION</subject></subj-group></article-categories><title-group><article-title>Прогнозирование максимальных напряжений в системе «вал–вкладыш» с помощью нейронной сети</article-title><trans-title-group xml:lang="en"><trans-title>Prediction of maximum stresses in the “shaft–insert” system using a neural network</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-3177-0959</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Боровков</surname><given-names>А. И.</given-names></name><name name-style="western" xml:lang="en"><surname>Borovkov</surname><given-names>A. I.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Боровков Алексей Иванович — кандидат технических наук, главный конструктор</p><p>Санкт-Петербург, 195251</p><p>sc 8840090300</p></bio><bio xml:lang="en"><p>Aleksei I. Borovkov — PhD, Chief Design Officer</p><p>Saint Petersburg, 195251</p><p>sc 8840090300</p></bio><email xlink:type="simple">vicerector.ap@spbstu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0008-2905-8448</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Карчевская</surname><given-names>А. С.</given-names></name><name name-style="western" xml:lang="en"><surname>Karchevskaia</surname><given-names>A. S.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Карчевская Анна Станиславовна — инженер</p><p>Санкт-Петербург, 195251</p></bio><bio xml:lang="en"><p>Anna S. Karchevskaia — Engineer</p><p>Saint Petersburg, 195251</p></bio><email xlink:type="simple">karchevskaya_as@spbstu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-5874-5994</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Новокшенов</surname><given-names>А. Д.</given-names></name><name name-style="western" xml:lang="en"><surname>Novokshenov</surname><given-names>A. D.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Новокшенов Алексей Дмитриевич — кандидат технических наук, доцент</p><p>Санкт-Петербург, 195251</p></bio><bio xml:lang="en"><p>Aleksei D. Novokshenov — PhD, Associate Professor</p><p>Saint Petersburg, 195251</p></bio><email xlink:type="simple">novoksh_ad@spbstu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0009-8189-0300</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Матвеева</surname><given-names>А. И.</given-names></name><name name-style="western" xml:lang="en"><surname>Matveeva</surname><given-names>A. I.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Матвеева Анастасия Игоревна — инженер</p><p>Санкт-Петербург, 195251</p></bio><bio xml:lang="en"><p>Anastasia I. Matveeva — Engineer</p><p>Saint Petersburg, 195251</p></bio><email xlink:type="simple">matveeva_ai@spbstu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-6404-6129</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Щербаков</surname><given-names>С. С.</given-names></name><name name-style="western" xml:lang="en"><surname>Sherbakov</surname><given-names>S. S.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Щербаков Сергей Сергеевич — доктор физико-математических наук, профессор, академик-секретарь Физико-технического отделения НАН Беларуси; главный научный сотрудник</p><p>Минск, 220072</p><p>sc 22036144400</p></bio><bio xml:lang="en"><p>Sergei S. Sherbakov — D.Sc. (Physics &amp; Mathematics), Professor, Academic Secretary of Physical and Mathematical Department, National Academy of Sciences of Belarus; Chief Researcher</p><p>Minsk, 220072</p><p>sc 22036144400</p></bio><email xlink:type="simple">sherbakovssr@yandex.by</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0001-7204-1974</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Климкович</surname><given-names>Н. М.</given-names></name><name name-style="western" xml:lang="en"><surname>Klimkovich</surname><given-names>N. M.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Климкович Никита Михайлович — старший преподаватель; младший научный сотрудник</p><p>Минск, 220030</p><p>Минск, 220072</p></bio><bio xml:lang="en"><p>Nikita M. Klimkovich — Senior Lecturer, Belarusian State University (BSU); Junior Researcher,</p><p>Minsk, 220030</p><p>Minsk, 220072</p></bio><email xlink:type="simple">Nikita.klimkovitch@yandex.ru</email><xref ref-type="aff" rid="aff-3"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0009-8988-6939</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Подгайская</surname><given-names>Д. А.</given-names></name><name name-style="western" xml:lang="en"><surname>Podgayskaya</surname><given-names>D. A.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Подгайская Дарья Александровна — заведующий лабораторией</p><p>Минск, 220072</p></bio><bio xml:lang="en"><p>Daria A. Podgayskaya — Head of Laboratory</p><p>Minsk, 220072</p></bio><email xlink:type="simple">dapodgayskaya@gmail.com</email><xref ref-type="aff" rid="aff-4"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-2701-2194</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Полещук</surname><given-names>М. М.</given-names></name><name name-style="western" xml:lang="en"><surname>Poleschuk</surname><given-names>M. M.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Полещук Михаил Михайлович — инженер-программист</p><p>Минск, 220030</p></bio><bio xml:lang="en"><p>Mikhail M. Poleschuk — Software Engineer</p><p>Minsk, 220030</p></bio><email xlink:type="simple">mikhailpaliashchuk@yandex.ru</email><xref ref-type="aff" rid="aff-5"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Санкт-Петербургский политехнический университет Петра Великого</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Peter the Great St. Petersburg Polytechnic University</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="ru"><institution>Национальная академия наук Беларуси; Объединенный институт машиностроения НАН Беларуси</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Physical and Mathematical Department, National Academy of Sciences of Belarus; The Joint Institute of Mechanical Engineering of the National Academy of Sciences of Belarus</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-3"><aff xml:lang="ru"><institution>Белорусский государственный университет; Объединенный институт машиностроения НАН Беларуси</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Belarusian State University (BSU); The Joint Institute of Mechanical Engineering of the National Academy of Sciences of Belarus</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-4"><aff xml:lang="ru"><institution>Объединенный институт машиностроения НАН Беларуси</institution><country>Россия</country></aff><aff xml:lang="en"><institution>The Joint Institute of Mechanical Engineering of the National Academy of Sciences of Belarus</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-5"><aff xml:lang="ru"><institution>Белорусский государственный университет</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Belarusian State University (BSU)</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2026</year></pub-date><pub-date pub-type="epub"><day>20</day><month>04</month><year>2026</year></pub-date><volume>26</volume><issue>2</issue><fpage>393</fpage><lpage>401</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Боровков А.И., Карчевская А.С., Новокшенов А.Д., Матвеева А.И., Щербаков С.С., Климкович Н.М., Подгайская Д.А., Полещук М.М., 2026</copyright-statement><copyright-year>2026</copyright-year><copyright-holder xml:lang="ru">Боровков А.И., Карчевская А.С., Новокшенов А.Д., Матвеева А.И., Щербаков С.С., Климкович Н.М., Подгайская Д.А., Полещук М.М.</copyright-holder><copyright-holder xml:lang="en">Borovkov A.I., Karchevskaia A.S., Novokshenov A.D., Matveeva A.I., Sherbakov S.S., Klimkovich N.M., Podgayskaya D.A., Poleschuk M.M.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://ntv.elpub.ru/jour/article/view/602">https://ntv.elpub.ru/jour/article/view/602</self-uri><abstract><p>Введение. Надежность работы машин во многом зависит от точности прогноза напряженно-деформированного состояния деталей трибофатических систем, особенно при высоких эксплуатационных нагрузках. Традиционные конечно-элементные расчеты обеспечивают высокую точность, однако характеризуются значительными вычислительными затратами и ограниченной возможностью быстрого варьирования параметров. В последние годы в инженерной практике все чаще применяются методы машинного обучения. Среди них нейронные сети представляют особый интерес, так как они позволяют описывать нелинейные зависимости между параметрами нагрузки и напряжениями, а также существенно ускоряют расчет по сравнению с традиционными моделями. Метод. Предложен подход к прогнозированию максимальных напряжений в системе «вал–вкладыш» с применением трехмерного конечно-элементного моделирования и последующего построения нейросетевой модели. Сформирована база данных, включающая результаты численных экспериментов при различных комбинациях изгибных и контактных нагрузок. Для обучения использовалась трехслойная полносвязная нейронная сеть с различными функциями активации каждого слоя. Выполнена оценка качества модели с помощью стандартных метрик Mean Squared Error, Mean Absolute Error и коэффициента детерминации R2. Основные результаты. Полученная нейронная сеть продемонстрировала высокую точность при прогнозировании максимальных напряжений как в вале, так и во вкладыше. Для обучающей выборки значение R2 составило 0,99991, а для тестовой — 0,99984, что подтверждает минимальные отклонения от данных конечно-элементных расчетов. Значение MAE составило менее 0,006, а максимальная относительная ошибка на тестовой выборке не превысила 3,2 %. Обсуждение. Разработанная нейросетевая модель показала способность к воспроизведению результатов конечно-элементного анализа для системы «вал–вкладыш», обеспечивая при этом значительное сокращение времени вычислений по сравнению с традиционным конечно-элементным моделированием. Модель построена для ограниченного диапазона нагрузок, поэтому дальнейшие исследования будут направлены на расширение исходного набора и включение новых материалов, что позволит оценить масштабируемость подхода и его устойчивость при усложнении условий.</p></abstract><trans-abstract xml:lang="en"><p>The reliability of machines largely depends on the accuracy of predicting the stress–strain state of components in tribofatigue systems, especially under high operating loads. Traditional finite element analysis provides high accuracy but requires significant computational resources and offers limited flexibility for rapid parameter variation. In recent years, machine learning methods have been increasingly applied in engineering practice. Among them, neural networks are of particular interest, as they allow nonlinear relationships between loads and stresses to be captured while significantly reducing computation time compared to traditional models. This work proposes an approach for predicting maximum stresses in the “shaft–insert” system by combining three-dimensional finite element modeling with subsequent neural network training. A database was created containing the results of numerical experiments for different combinations of bending and contact loads. A fully connected neural network with three hidden layers and different activation functions was used for training. The quality of the model was assessed using standard metrics: Mean Squared Error, Mean Absolute Error (MAE), and the coefficient of determination R2. The trained neural network demonstrated high accuracy in predicting maximum stresses both in the shaft and in the insert. For the training set, the R2 value reached 0.99991, and for the test set it was 0.99984, confirming minimal deviations from finite element results. The MAE was less than 0.006, while the maximum relative error in the test set did not exceed 3.2 %. The developed neural network model demonstrated the ability to reproduce the results of finite element analysis for the “shaft–insert” system while providing a substantial reduction in computation time compared to traditional finite element simulations. The model was constructed for a limited range of loads; therefore, further research should focus on expanding the dataset and including additional materials, which will make it possible to evaluate the scalability of the approach and its robustness under more complex conditions.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>нейронные сети</kwd><kwd>машинное обучение</kwd><kwd>трибофатическая система</kwd><kwd>максимальные напряжения</kwd><kwd>напряженно-деформированное состояние</kwd><kwd>чугун «МОНИКА»</kwd><kwd>механика твердого тела</kwd><kwd>конечно-элементное моделирование</kwd><kwd>контактное взаимодействие</kwd></kwd-group><kwd-group xml:lang="en"><kwd>neural networks</kwd><kwd>machine learning</kwd><kwd>tribo-fatigue system</kwd><kwd>stress–strain state</kwd><kwd>maximum stresses</kwd><kwd>MONICA</kwd><kwd>solid mechanics</kwd><kwd>finite element modeling</kwd><kwd>contact interaction</kwd></kwd-group><funding-group><funding-statement xml:lang="ru">Работа выполнена при поддержке Российского научного фонда, проект № 23-РБ-09-27 от 15.12.2023, а также при поддержке Белорусского республиканского фонда фундаментальных исследований (проект № Т24СПбГ-003).</funding-statement><funding-statement xml:lang="en">This work was supported by the Russian Science Foundation (project No. 23-RB-09-27, 15.12.2023) and the Belarusian Republican Foundation for Fundamental Research (project No. T24SPbG-003).</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Sosnovskiy L.A. 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